K-moduli of log del Pezzo pairs and variations of GIT

Abstract

We study the K-moduli of log del Pezzo pairs formed by a del Pezzo surface of degree $d$ and an anti-canonical divisor. These moduli spaces naturally depend on one parameter, providing a natural problem in variations of K-moduli spaces. For degrees 2, 3, 4, we establish an isomorphism between the K-moduli spaces and variations of Geometric Invariant Theory compactifications, which generalizes the isomorphisms in the absolute cases established by Odaka--Spotti--Sun and Mabuchi--Mukai.

arXiv comment46 pages. Rewrote part of Section 4, added Lemma 4.6, expanded and clarified 4.22 and 4.23, and corrected an error in 4.25